How to find the eccentricity of the earth's orbit. Orbits of the planets of the solar system. How to remember all the planets
Known three cyclic processes, leading to slow, so-called secular fluctuations in the values of the solar constant. Corresponding secular climate changes are usually associated with these fluctuations in the solar constant, which was reflected in the works of M.V. Lomonosov, A.I. Voeykova and others. Later, when developing this issue, arose astronomical hypothesis of M. Milankovitch, explaining changes in the Earth's climate in the geological past. Secular fluctuations of the solar constant are associated with slow changes in the shape and position of the earth's orbit, as well as the orientation of the earth's axis in world space, caused by the mutual attraction of the earth and other planets. Since the masses of the other planets of the Solar System are significantly less than the mass of the Sun, their influence is felt in the form of small perturbations of the elements of the Earth’s orbit. As a result of the complex interaction of gravitational forces, the path of the Earth around the Sun is not a constant ellipse, but a rather complex closed curve. The irradiation of the Earth following this curve is continuously changing.
The first cyclic process is change in orbital shape from elliptical to almost circular with a period of about 100,000 years; it is called eccentricity oscillation. Eccentricity characterizes the elongation of the ellipse (small eccentricity – round orbit, large eccentricity – orbit – elongated ellipse). Estimates show that the characteristic time of change in eccentricity is 10 5 years (100,000 years).
Rice. 3.1 − Change in Earth's orbital eccentricity (not to scale) (from J. Silver, 2009)
Changes in eccentricity are non-periodic. They fluctuate around the value of 0.028, ranging from 0.0163 to 0.0658. Currently, the orbital eccentricity of 0.0167 continues to decrease, and its minimum value will be reached in 25 thousand years. Longer periods of decrease in eccentricity are also expected - up to 400 thousand years. A change in the eccentricity of the earth's orbit leads to a change in the distance between the Earth and the Sun, and, consequently, in the amount of energy supplied per unit time to a unit area perpendicular to the sun's rays at the upper boundary of the atmosphere. It was found that when the eccentricity changes from 0.0007 to 0.0658, the difference between the solar energy fluxes from the eccentricity for cases when the Earth passes the perihelion and aphelion of the orbit changes from 7 to 20−26% of the solar constant. Currently, the Earth's orbit is slightly elliptical and the difference in solar energy flux is about 7%. During the greatest ellipticity, this difference can reach 20−26%. It follows from this that at small eccentricities the amount of solar energy arriving at the Earth, located at perihelion (147 million km) or aphelion (152 million km) of the orbit, differs slightly. At the greatest eccentricity, more energy comes to perihelion than to aphelion by an amount equal to a quarter of the solar constant. The following characteristic periods are identified in eccentricity fluctuations: about 0.1; 0.425 and 1.2 million years.
The second cyclic process is a change in the inclination of the earth's axis to the ecliptic plane, which has a period of about 41,000 years. During this time, the slope changes from 22.5° (21.1) to 24.5° (Fig. 3.2). Currently it is 23°26"30". An increase in the angle leads to an increase in the height of the Sun in summer and a decrease in winter. At the same time, insolation will increase in high latitudes, and at the equator it will decrease slightly. The smaller this inclination, the smaller the difference between winter and In the summer, warmer winters are snowier, and colder summers prevent all the snow from melting. Snow accumulates on the Earth, promoting the growth of glaciers. As the slope increases, the seasons are more pronounced, winters are colder and there is less snow, and summers are warmer and there is more snow and ice. melts. This promotes the retreat of glaciers to the polar regions. Thus, the increase in angle increases seasonal, but reduces latitudinal differences in the amount of solar radiation on Earth.
Rice. 3.2 – Change in the inclination of the Earth's rotation axis over time (from J. Silver, 2009)
The third cyclic process is the oscillation of the axis of rotation of the globe, called precession. Precession of the earth's axis- This is the slow movement of the Earth's rotation axis along a circular cone. The change in the orientation of the earth's axis in world space is due to the discrepancy between the center of the earth, due to its oblateness, and the gravitational axis of the earth–moon–sun. As a result, the Earth's axis describes a certain conical surface (Fig. 3.3). The period of this oscillation is about 26,000 years.
Rice. 3.3 – Precession of the Earth’s orbit
Currently, the Earth is closer to the Sun in January than in June. But due to precession, after 13,000 years it will be closer to the Sun in June than in January. This will lead to an increase in seasonal temperature variations in the Northern Hemisphere. The precession of the earth's axis leads to a mutual change in the position of the winter and summer solstice points relative to the perihelion of the orbit. The period with which the mutual position of the orbital perihelion and the winter solstice point repeats is equal to 21 thousand years. More recently, in 1250, the perihelion of the orbit coincided with the winter solstice. The Earth now passes perihelion on January 4th, and the winter solstice occurs on December 22nd. The difference between them is 13 days, or 12º65". The next coincidence of the perihelion with the winter solstice point will occur after 20 thousand years, and the previous one was 22 thousand years ago. However, between these events the summer solstice point coincided with the perihelion.
At small eccentricities, the position of the summer and winter solstices relative to the orbital perihelion does not lead to a significant change in the amount of heat entering the earth during the winter and summer seasons. The picture changes dramatically if the orbital eccentricity turns out to be large, for example 0.06. This is how the eccentricity was 230 thousand years ago and will be in 620 thousand years. At large eccentricities of the Earth, the part of the orbit adjacent to the perihelion, where the amount of solar energy is greatest, passes quickly, and the remaining part of the elongated orbit through the vernal equinox to the aphelion passes slowly, for a long time being at a great distance from the Sun. If at this time the perihelion and the winter solstice point coincide, the Northern Hemisphere will experience a short, warm winter and a long, cool summer, while the Southern Hemisphere will experience a short, warm summer and a long, cold winter. If the summer solstice point coincides with the perihelion of the orbit, then hot summers and long cold winters will be observed in the Northern Hemisphere, and vice versa in the Southern Hemisphere. Long, cool, wet summers are favorable for the growth of glaciers in the hemisphere where most of the land is concentrated.
Thus, all of the listed different-sized fluctuations in solar radiation are superimposed on each other and give a complex secular course of changes in the solar constant, and, consequently, a significant impact on the conditions for climate formation through changes in the amount of solar radiation received. Fluctuations in solar heat are most pronounced when all three of these cyclic processes are in phase. Then great glaciations or complete melting of glaciers on Earth are possible.
A detailed theoretical description of the mechanisms of influence of astronomical cycles on the earth's climate was proposed in the first half of the 20th century. the outstanding Serbian astronomer and geophysicist Milutin Milankovic, who developed the theory of the periodicity of ice ages. Milankovitch hypothesized that cyclic changes in the eccentricity of the Earth's orbit (its ellipticity), fluctuations in the angle of inclination of the planet's rotation axis and the precession of this axis can cause significant changes in the climate on Earth. For example, about 23 million years ago, the periods of the minimum value of the eccentricity of the Earth's orbit and the minimum change in the inclination of the Earth's rotation axis coincided (it is this inclination that is responsible for the change of seasons). For 200 thousand years, seasonal climate changes on Earth were minimal, since the Earth's orbit was almost circular, and the tilt of the Earth's axis remained almost unchanged. As a result, the difference in summer and winter temperatures at the poles was only a few degrees, the ice did not have time to melt over the summer, and there was a noticeable increase in its area.
Milankovitch's theory has been repeatedly criticized, since variations in radiation for these reasons relatively small, and doubts were expressed whether such small changes in high-latitude radiation could cause significant climate fluctuations and lead to glaciations. In the second half of the 20th century. A significant amount of new evidence has been obtained about global climate fluctuations in the Pleistocene. A significant proportion of them are columns of oceanic sediments, which have an important advantage over terrestrial sediments in that they have a much greater integrity of the sequence of sediments than on land, where sediments have often been displaced in space and repeatedly redeposited. Spectral analysis of such oceanic sequences dating back to the last approximately 500 thousand years was then carried out. Two cores from the central Indian Ocean between the subtropical convergence and the Antarctic oceanic polar front (43–46°S) were selected for analysis. This area is equally far from the continents and therefore is little affected by fluctuations in erosion processes on them. At the same time, the area is characterized by a fairly high rate of sedimentation (more than 3 cm/1000 years), so that climatic fluctuations with a period significantly less than 20 thousand years can be distinguished. As indicators of climate fluctuations, we selected the relative content of the heavy oxygen isotope δO 18 in planktonic foraminifera, the species composition of radiolarian communities, as well as the relative content (in percentage) of one of the radiolarian species Cycladophora davisiana. The first indicator reflects changes in the isotopic composition of ocean water associated with the emergence and melting of ice sheets in the Northern Hemisphere. The second indicator shows past fluctuations in surface water temperature (T s) . The third indicator is insensitive to temperature, but sensitive to salinity. The vibration spectra of each of the three indicators show the presence of three peaks (Fig. 3.4). The largest peak occurs at approximately 100 thousand years, the second largest at 42 thousand years, and the third at 23 thousand years. The first of these periods is very close to the period of change in the orbital eccentricity, and the phases of the changes coincide. The second period of fluctuations in climate indicators coincides with the period of changes in the angle of inclination of the earth's axis. In this case, a constant phase relationship is maintained. Finally, the third period corresponds to quasiperiodic changes in precession.
Rice. 3.4. Oscillation spectra of some astronomical parameters:
1 - axis tilt, 2 - precession ( A); insolation at 55° south. w. in winter ( b) and 60° N. w. in summer ( V), as well as the spectra of changes in three selected climate indicators over the last 468 thousand years (Hays J.D., Imbrie J., Shackleton N.J., 1976)
All this makes us consider changes in the parameters of the earth’s orbit and the tilt of the earth’s axis to be important factors in climate change and indicates the triumph of Milankovitch’s astronomical theory. Ultimately, global climate fluctuations in the Pleistocene can be explained precisely by these changes (Monin A.S., Shishkov Yu.A., 1979).
I. Kulik, I.V. Sandpiper
Method for determining the eccentricity of a planet's orbit
Key words: time, orbit, apsidal line, parameter line, mean anomaly, true anomaly, center equation, time ray.
V.I. Kulik, I.V. Kulik
Technique of definition of eccentricity of an orbit of the planet
The technique of defining eccentricity orbits only by measurement of angular position of a planet is offered.
Keywords: time, orbit, the line of apses, the line parameters, mean anomaly, the true anomaly, the equation of the center, evenly rotating beam time.
There are various expressions for determining the orbital eccentricity.
Here are a series of expressions for determining the eccentricity "e" of the orbit.
Rice. 1. When moving from RB to RH, with c = 1.5; A = 4.5; Ro = 4 if if ¥ = ^, then< = 1,230959418 5. e = VH - VB VH + VB R B - RH RB + RH However, almost all expressions contain linear ones. In theoretical astronomy, the relationship is considered parameters that, while on Earth, can be measured between the true anomaly φ and the average anomaly % directly impossible. Parameters of the planet's orbit. In the Earth's orbital movement, see Fig. 2, (Fig. 1). We pursue the goal of determining the true anomaly of the Earth’s position in orbit The eccentricity of any planetary system, measured by the angle φ between the radius vectors: the Sun only its angular position on the celestial sphere and (focus of the orbit M) - perihelion and the Sun - Earth, i.e. the period of its revolution around the center. Rice. 2. Orbit parameters The average anomaly is the angle between the radius vector Sun - perihelion (on the apsidal line) and the radius vector (not shown in Fig. 2), uniformly rotating (in the direction of the Earth's movement) with angular velocity n = , where T is the period the revolution of the Earth around the Sun, expressed in solar (average) units. Moreover, the rotation of the vector (Sun M - Earth t) occurs in such a way that its end, located in orbit and moving unevenly along it, simultaneously with the end of the vector uniformly rotating (in the direction of the Earth’s movement) with angular velocity n = ■ passes the apse points, that is, for apsidal points we have φ = £. With a value n, the average anomaly is determined by the formula: * / 2 - n. where t is the time interval from the moment of passage Earth through perihelion. Difference φ - £ = φ---1 = P is called the equation of the center. It reflects the unevenness of the Earth's annual movement; this applies to the same extent to the apparent annual movement of the Sun. In theoretical astronomy, the formula for this difference is approximately determined. In the perigee region (PE) the movement of the planet is fast, and in the apogee region (AP) it is slow. In the section of the trajectory between PE and AP, the radius vector of the Earth’s revolution moves ahead of the uniformly rotating ray of time, i.e., angle p > C (Fig. 3), while on the other half of the orbit, or on the other side of apsidal lines, between points AP and PE, the radius vector of the Earth’s revolution moves behind the uniformly rotating ray of time, i.e. angle p< С (Fig. 3). In Fig. Figure 3 also shows the transfer of the origin of motion from perigee t. O on the line of apses to t. Og (t.) on the line of equinoxes. And if we count time (and other parameters) from the line of apses (whether from the point PE a new natural cycle of movement began or from the point AP), then the calculations show the symmetry of all parameters, see the graph f relative to the line sd. But if we shift the reference point to the line of equinoxes in point Og (in point G2) (Fig. 3), then the symmetry is destroyed, see the graph of φ "relative to line C, see Fig. 3. Just like the graph of the angle p" , and the graph of the angle T] is not symmetrical relative to line C". Only in the area indicated by arrows B, the planet “overtakes” time and angle p" > C, at all other points of the trajectory the planet “lags behind” the uniformly rotating ray of time and angle (< д (рис. 3). The graph of the angle of ascension of the Sun, angle /, is always considered between the points of the spring and autumn equinox, i.e. between points y and O on the line equinoxes, it is similar relative to line C (or time lines?" = С "р), however, the duration of time (i.e., depending on time) is different on both sides of the line of equinoxes (Fig. 2 and 3). Rice. 3. Change of reference point: O - from perigee, O" - from the line of equinoxes The orbital eccentricity can be determined from the equation for the planet's mean anomaly, namely: Explanation of the proposed formula (*) when moving from apogee (AP): where = 2 arcSin J^1 * e^ zA ; whence z^ = Sin2^. In turn, the value of zA depends on the angle fA or za =~l-~-, whence the true anomaly planets: (a = arcCoS Explanation of the proposed formula (*) when moving from perigee (PE): %п =^f- fn =^п - e sinvnl ¥ zn -eK.-e)J¿) where ШП = 2 arcSin J--- zп, whence zП = -2- Sin2 ^П- In turn, the value of 2P depends on the FP angle or Zп (1- cos(n) 1 + e cos rn where does the true anomaly come from? planets: rp = arcCoS Further. Figures 4 and 5 show the orbits of a planet that have the same average distance A from the center around which the planet revolves. In addition, in Fig. 4, the orbits are shown with a fixed (fixed) center of symmetry at point O and a variable position of the focus (/1, /2,/3) of the orbit, and in Fig. 5, the orbits are shown with a stationary (fixed) position of the focus at point ^ and a variable position of the center of symmetry (point Oz, O2, Oz), orbits. Radius Yao is an orbital parameter (Fig. 2). In the above formula (*), the sign (+) corresponds to the case when the beginning of the movement from apogee to perigee is taken as the origin of reference or movement, that is, from the radius Jav (or Jaap) to the radius Yang (or Jape), and the sign (-) corresponds to the case when the beginning of the reference or movement is taken to be the beginning of movement from perigee to apogee, that is, from the radius Yang (or Yape) to the radius Yav (or Jaap). Rice. 4. Orbital parameters for a fixed center of symmetry O Rice. 5. Orbit parameters with fixed focus F If we consider, Fig. 2, 4 and 5, when the planet moves from the apogee (from the radius Rav) to the angle (a = Ra = , (and before (a = 2~ " - the planet is approaching the center of mass (to the focus of the orbit) and formula (1) is simplified, then time will pass: arcSin^1 + e) + e-y/1 - e2 or tB = tA = If we consider, Fig. 2, 4 and 5, when the planet moves from perigee (from the radius Yang) at an angle Рн = Рп = 2", then is, - movement from the angle (n = 0 to Pn =, - the planet moves away from the center of mass (from the focus of the orbit) and formula (2) is simplified, - then time will pass: or tH = tn = - Then the average anomaly of the planet as the planet moves from the apogee will be: = "tA =¥a + e - sin^A = 2 arcSinу" (1 + e) E - jre = 2 - arcSin + e-JR0 . 2 V2 - A V A Here we have everywhere: (a = Рп = , и = 1п = 0. Accordingly, the average anomaly of the planet when the planet moves from perigee will be: Tn =Wu - e - sin^n = 2 - arcSin - e-^l 1 - e2 = 2 - arcSin^^-. If we now consider two simplified formulas, namely: Dr - tA = 2 - arcSin Aii+^i + e-V 1 - e2 Tn = 2 - arcSin J- e-VI-\ then in each of them, in addition to the orbital period T, supposedly two more unknown quantities are visible: u and e. But this is not so. From astronomical observations we can always determine: 1) the period of revolution of the planet - T; 2) angle Рд = Рп = - rotation of the ray along which the planet moves; 3) time tA or for which the specified beam will rotate through an angle p^ = rd = rts = - from the apsidal line. If the sidereal period of revolution of the planet is T = 31558149.54 seconds, and the ray on which the planet is located rotates through the angle рг- = рА = -, and at the same time, the time interval from the moment the Earth passes through the apogee apsidal lines, or time tA of the planet’s movement from apogee to angle p = - is the quantity g = T.0.802147380127504 = 8057787.80589431 [s], p then from the transcendental equation GA = ^T. 0.802147380127504 ^ = = 2.0.802147380127504 = 1. 6042947602 5501= 2. arcW^1^ + e ^ 1_ e2, or 0.802147380127504[rad] = arcBt^1^ +£^ 1 _e2, determine the eccentricity. The eccentricity value is equal to e = 0.01675000000. Similarly, if the time interval from the moment the Earth passes through the perigee of the apsidal line, or the time ^ of the planet’s movement from perigee to an angle p = F is the value GP = T. 0.768648946667393 = 7721286.96410569 [s], then from 2 p transcendental equation GP = -.(T. 0.768648946667393 bp t p t I p 2-0.768648946667393 = 1.53729789333479 = 2 arcSini^-^ _1 _e2 or 0.768648946667393 = a^t^-^ _£1 _e2, the orbital eccentricity can be determined. The eccentricity value is equal to e = Here + £д = 1.6042947602550 + 1.53729789333479: 0.016750000. 3.14159265358979 = p. Here always fl + fp = p. Here always It is clear that this problem is reversible, and using two other known quantities one can always find ^ + t^ = - unknown third quantity. Literature 1. Kulik V.I. Organization of planets in the solar system. Structural organization and oscillatory motions of planetary systems in a multi-mass solar system / V.I. Kulik, I.V. Kulik // Verlag. - Deutschland: Lap lambert Academic Publishing, 2014. - 428 p. 2. Mikhailov A.A. The Earth and its rotation. - M.: Nauka, 1984. 3. Khalkhunov V.Z. Spherical astronomy. - M.: Nedra, 1972. - 304 p. If the orbit is not a circle, then the speed of the planet's rotation around the Sun and, therefore, the rate of change of deviation will not be constant. They change faster near perihelion and more smoothly near apogee. Let's enter the value C (deg), which will show the difference between the true deviation during the given 24 hours and the average deviation value (0 -<0>= C°). The quantity C is called the equation of the center (an ancient name). Since the Earth's rotation speed around its axis is 1° per 4 minutes, the time between actual noon and mean noon, depending on the orbital eccentricity, can be determined as EOTt = 4C. (61) In the equation mentioned above, C is taken in degrees. The term responsible for the influence of eccentricity in the equation of time changes throughout the year according to a sinusoidal law, becoming zero at apogee and perigee. The maxima of this term are shifted by 8 minutes relative to the center of the interval between apogee and perigee. This dependence is shown in Fig. 10.17. The value of C for any point in time can be determined using 0, which is found by equation (44) by subtracting the average deviation, which is calculated by equation (60). In many cases, it is much easier to determine C using the empirical equation below (see http://www.srrb.noaa.gov/highlights/sunrise/program.txt): (9) = 357.529 11 + 35 999.050 29T - 0.000153 IT2; (62 > C = (l, 914 602 - 0.004 817Г - 0.000 014Г2) sin (0) + + (0.019 993 - 0.000101Г) sin (2 (0)) + 0.000 289 sin (3 (0)) - (63) Orbital inclination If the orbit were a circle, but its inclination was not zero, then, despite the constant rate of change zi. - liptic longitude, the rate of change of right ascension will not be constant. But, as you know, the zenith of the Sun depends precisely on right ascension. The day after the spring equinox, the meaning of right ascension 9? = arctg (cost tgA) = arctg (cos (23.44") tg (0.985 647" jj = Arctg(0.91747 -0.017 204) = arctg(0.015785),<64) 9^ = 0.904322e. (65) In order for the Sun to be at its zenith, it is necessary that the Earth rotated an additional 0.904322°, rather than 0.985647°, which corresponds to a zero orbital inclination. That is, noon will come earlier than in the absence of orbital inclination. The difference will be 4(0.985 674 - 0.904 322) = = 0.325 min. In general The term in the equation of time that depends on the inclination, as well as the term that depends on the eccentricity, will vary according to a sinusoidal law. However, the term that depends on the orbital inclination will have two maxima during the year. The zeros will fall on the days of the equinox and solstices, and not on the moments of apogee and perigee. The amplitude of EOTobhq is 10 min. The behavior of this function is shown in Fig. 10.18. and the behavior of the general equation of time EOT, which is the sum of EOTsssSH and EOTobliq, is shown in Fig. 10.19. It is important not to confuse concepts such as apogee and perigee, which define the points closest to the Sun and farthest from it in the Earth’s orbit, with solstice days, which occur when the Sun’s declination is extreme (5 = + 23.44°). Sometimes it happens that the days of perigee and apogee coincide with the days of the solstices, but these are nothing more than random coincidences. Typically, the difference between the dates of apogee and summer solstice is about 12 days. Approximately the same time interval is observed between perigee and the winter solstice (Table 10.5). 10.1. Let a certain traveler find himself in some unknown place on Earth at an unknown time of year. Due to the constant night clouds, he is not able to navigate by the stars, but he can quite accurately determine the time of sunrise and the length of his shadow at noon. Sunrise occurs at 05:20 local time, and the length of the shadow at noon is 1.5 times its height. Determine the day of the year and the latitude of the area. Is there a unique solution to the problem? 10.2. The tourist has an accurate electronic watch, with the help of which he determined that 10 hours 49 minutes and 12 seconds pass between sunrise and sunset. He knows the date - January 1, 1997. Help him find the latitude of the place where he is. 10.3. The windows of a building in Palo Alto, California, USA (latitude 37.4° N) are oriented south-southeast. During what period of the year does the sun's rays enter the room during sunrise? Neglect the size of the solar disk and the shading of the sun. What time does the sun rise on the first and last day of this period? What is the solar radiation flux density on a wall with the same orientation at noon on the equinoxes? 10.4. Consider an ideal focusing concentrator. An increase in the degree of concentration leads to an increase in temperature to a certain limit. Determine the maximum achievable degree of concentration under Mars conditions for 2-D and 3-O concentrators. The orbital radius of Mars is 1.6 AU. e., 1 a. e. = 150 million km. The angular diameter of the sun is 0.5°. 10.5. Let some distribution function have the form ABOUT? = f _ i. f1 df J 2 Determine at what value / this function has a maximum. Plot the d/yd/in Function from/for the interval in which dP/df > 0. Now enter a new variable X = c/f where c is a certain constant. At what value / does the dP/dX function have a maximum. Plot a graph | dP/d X | as a function of / The expedition begins work on Mars on November 15, 2007, corresponding to the 118th Martian day of the year. The expedition ends up on Mars at a point with coordinates 17° N. w. and 122° east. d. at the moment of sunrise. The five-person expedition must use the day to get the equipment they need to survive the cold night into operation. Previously, before the arrival of the expedition, an installation was installed with the help of robots that made it possible to extract water from rock hydrates using concentrated solar radiation. Evaluate the daily need for will for yourself. Electricity is planned to be generated using photovoltaic converters (PVCs) and accumulated in hydrogen and oxygen obtained from water by electrolysis. The efficiency of photoconverters is 16.5% with one “Mars sun”. Hubs do not apply. PV panels are located horizontally on the surface of Mars. Electrolyzer efficiency is 95%. The inclination of the equatorial plane of Mars to the plane of its orbit is 25.20°. The average daily temperature on the surface of Mars is 300 K (slightly higher than on Earth, where it is 295 K). The Martian night, however, is much colder1 The average night temperature is 170 K (on Earth - 275 K). Let us assume (although this is not true) that the spring equinox falls on the 213th day from the beginning of the year. Define the Martian hour It as 1/24 of the average annual period between* the corresponding sunrises. 1. What is the duration of the sunny day on the day the expedition arrives? 2. Calculate the insolation of the horizontal surface (W/m2) average for d i Martian days, lasting (h) 24hm. 3. Estimate the oxygen consumption of five astronauts based on the fact that they need 2500 kcal per Martian day. Assume that the energy consumption mechanism is exclusively related to glucose, for which the enthalpy of “burning” is equal to 16 MJ/kg. 4. How much energy will be required to obtain the required amount of water by electrolysis? 5. What should be the area of solar panels to ensure the required oxygen production? 6. Assume that the temperature in the room with the astronauts is equal to the average temperature on the surface of Mars and that the temperature of the “air” on Mars changes from 300 K at noon to 175 K at midnight and vice versa. The astronauts live in a plastic hemisphere with a diameter of 10 m. The thermal resistance of the capsule wall is 2 m2 K W1. Heat losses through the floor can be neglected. Inside the living capsule, the temperature is maintained at 300 K, and with steam; is equal to 175 K. Assume that the thermal emissivity coefficient of the outer surface of the capsule is 0.5. What are the daily hydrogen requirements? What is the required area of solar panels? 10.7. What was the length of the shadow from a 10-meter tree in Palo Alto, USA, on March 20, 1991 at 2 p.m.? Estimate to within 20 cm. 10.8. Calculate the optimal azimuth of the vertical surface to ensure maximum average annual solar radiation collection under the following conditions. The surface is at a latitude of 40° N. w. in an area where every morning until 10:00 a.m. there is dense fog that does not allow solar radiation to reach the Earth’s surface, and the rest of the day the sky is clear. Compare the resulting insolation with insolation on a horizontal surface located at the equator 10.10. What is the insolation (W/m2) on a surface facing due east, with an angle of inclination to the horizon of 25° in a place with a latitude of 45° N. w. at 10:00 on April 1? 10.11. What is the azimuth of the Sun at sunset on the summer solstice at latitude 50° N. sh.? 10.12. The photovoltaic battery has an efficiency of 16.7%. It is located in a place located at a latitude of 45° N. w. Observations are carried out on April 1, 1995 at 10:00 am. If the photobattery is oriented strictly towards the Sun, its power will be equal to 870 W. What power will the same battery produce if it is installed due east with an angle of inclination to the horizon of 25°? 10.13. Let's consider a vapor compression heat pump, the efficiency of which is 0.5 as the maximum possible. The heat pump consumes mechanical power to drive the compressor W, as a result of which the thermal power Qc is taken from the outside air with a temperature of -10 °C and the heat flow QA = Qc+ W is sent to the heated room with a temperature of 25 °C. Calculate the conversion coefficient of the heat pump, equal to the ratio useful thermal power to mechanical power. 10.14. The minimum zenith angle of the Sun on January 1, 2000 is 32.3°. At this moment in time it was located strictly south of the observer. Determine the latitude of the observer's position. 10.15. A certain airplane is used as a radio repeater. It is equipped with 14 electric generators with a capacity of 1.5 kW and runs at a speed of 40 km/h at an altitude of 30 km. The wingspan is 75.3 m. The maximum power of the photovoltaic battery placed on the wings is 32 kW with a perpendicular incidence of solar radiation on them. 1. What is the distance to the geometric horizon visible from the flight altitude? Note that the geometric horizon differs from the radio horizon, which< - рый существенно превышает первый из-за особенностей распространения радиоволн в атмосфере. 2. What is the area of direct coverage of the earth's surface from an aircraft? 3. Let the airplane fly over an area located at 37.8° N. w. Determine the minimum duration of daytime sunshine during the year at the height of the apparatus. 4. What is the average daily insolation on solar cells located on the wings in a horizontal plane on the day discussed above? Since the apparatus is located above the clouds, it can be assumed that the intensity of solar radiation at this altitude (solar constant) is 1200 W/m2. 5. Assume that the efficiency of solar cells is 20%, and the efficiency of the processes of accumulating and using electricity is equal to unity. The total power consumed by the aircraft, required both for maintaining the flight and for relaying, is 10 kW. To simplify the problem, the wings of the glider can be considered rectangular. FEP located l. on 90% of the wing surface. What should be the chord (width) of the wing to ensure the functionality of the considered flying retrans. ■ torus? Eccentricity (denoted e or ε) is one of the six Keplerian orbital elements. Along with the semimajor axis, it determines the shape of the orbit. Kepler's first law states that the orbit of any planet in the solar system is an ellipse. Eccentricity determines how different the orbit is from a circle. It is equal to the ratio of the distance from the center of the ellipse (c) to its semimajor axis focus (a). The circle's focus coincides with the center, i.e. c = 0. Also any ellipse c Orbit of Sedna. At the center of coordinates is the Solar System, surrounded by a swarm of planets and known Kuiper Belt objects. In our system, the orbits of the planets are unremarkable. It has the most “circular” orbit. Its aphelion is only 1.4 million km greater than perihelion, and its eccentricity is 0.007 (for the Earth it is 0.016). Pluto moves in a rather elongated orbit. With ε = 0.244, it sometimes approaches the Sun even closer than Neptune. However, since Pluto recently fell into the category of dwarf planets, Mercury now has the most elongated orbit among the planets, with ε = 0.204. Among the dwarf planets, Sedna is the most notable. Having ε = 0.86, it makes a full revolution around the Sun in almost 12 thousand years, moving away from it at aphelion by more than a thousand astronomical units. However, even this is incomparable with the orbital parameters of long-period comets. Their orbital periods sometimes amount to millions of years, and many of them will never return to the Sun at all - i.e. have an eccentricity greater than 1. can contain trillions of comets distant from the Sun by 50-100 thousand astronomical units (0.5 - 1 light years). At such distances they can be influenced by other stars and galactic tidal forces. Therefore, such comets can have very unpredictable and variable orbits with very different eccentricities. Finally, the most interesting thing is that even the Sun does not have a circular orbit at all, as it might seem at first glance. As is known, the Sun moves around the center of the Galaxy, making its way in 223 million years. Moreover, due to countless interactions with stars, it received a rather noticeable eccentricity of 0.36. Comparison of the orbit of HD 80606 b with the inner planets of the Solar System The discovery of other solar systems inevitably entails the discovery of planets with very bizarre orbital parameters. An example of this is the eccentric Jupiters, gas giants with fairly high eccentricities. In systems with such planets, the existence of planets similar to Earth is impossible. They will inevitably fall on the giants or become their satellites. Among the eccentric Jupiters discovered so far, HD 80606b has the highest eccentricity. It moves around a star slightly smaller than our Sun. This planet at perihelion approaches the star 10 times closer than Mercury does to the Sun, while at aphelion it moves away from it by almost an astronomical unit. Thus, it has an eccentricity of 0.933. It is worth noting that although this planet crosses the zone of life, there can be no talk of any types of the usual biosphere. Its orbit creates an extreme climate on the planet. During a short period of approach to the star, the temperature of its atmosphere changes by hundreds of degrees in a matter of hours, resulting in wind speeds reaching many kilometers per second. Other planets with high coefficients have similar conditions. The same, for example, when approaching the Sun, acquires an extensive atmosphere, which settles in the form of snow as it moves away. At the same time, all Earth-like planets have orbits close to circular. Therefore, eccentricity can be called one of the parameters that determines the possibility of the presence of organic life on the planet.
The corresponding ellipse. More generally, the orbit of a celestial body is a conic section (that is, an ellipse, parabola, hyperbola, or straight line), and it has an eccentricity. Eccentricity is invariant under plane motions and similarity transformations. Eccentricity characterizes the “compression” of the orbit. It is calculated by the formula: The appearance of the orbit can be divided into five groups: The table below shows the orbital eccentricities for some celestial bodies (sorted by the size of the semi-major axis of the orbit, satellites - indented).
Determination of eccentricity
Eccentricities of Solar System objects
Eccentricities in other systems
texvc not found; See math/README for setup help.): \varepsilon = \sqrt(1 - \frac(b^2)(a^2)), Where Unable to parse expression (Executable file texvc not found; See math/README for setup help.): b- semi-minor axis, Unable to parse expression (Executable file texvc not found; See math/README for setup help.): a- major axle shaft texvc not found; See math/README for setup help.): \varepsilon = 0- circumferencetexvc not found; See math/README for setup help.): 0< \varepsilon < 1
- ellipsetexvc not found; See math/README for setup help.): \varepsilon = 1- parabolatexvc not found; See math/README for setup help.): 1< \varepsilon < \infty
- hyperboletexvc not found; See math/README for setup help.): \varepsilon = \infty- direct (degenerate case)Heavenly body
Orbital eccentricity
Mercury
0,205
0.205
Venus
0,007
0.007
Earth
0,017
0.017
Moon
0,05490
0.0549
(3200) Phaeton
0,8898
0.8898
Mars
0,094
0.094
Jupiter
0,049
0.049
And about
0,004
0.004
Europe
0,009
0.009
Ganymede
0,002
0.002
Callisto
0,007
0.007
Saturn
0,057
0.057
Titanium
0,029
0.029
Halley's Comet
0,967
0.967
Uranus
0,046
0.046
Neptune
0,011
0.011
Nereid
0,7512
0.7512
Pluto
0,244
0.244
Haumea
0,1902
0.1902
Makemake
0,1549
0.1549
Eris
0,4415
0.4415
Sedna
0,85245
0.85245
see also
Write a review about the article "Orbital eccentricity"
Notes
View this template
Laws and tasks
Celestial sphere
Orbit parameters
Movement
celestial bodies
Astrodynamics
Space flight
Spacecraft orbits
An excerpt characterizing the eccentricity of the orbit
My legs gave way from horror, but for some reason Karaffa did not notice this. He glared at my face with a flaming gaze, not answering and not noticing anything around. I couldn’t understand what was happening, and this whole dangerous comedy frightened me more and more... But then something completely unexpected happened, something completely outside the usual framework... Caraffa came very close to me, that’s all also, without taking his burning eyes off, and almost without breathing, he whispered:
– You cannot be from God... You are too beautiful! You are a witch!!! A woman has no right to be so beautiful! You are from the Devil!..
And turning around, he rushed out of the house without looking back, as if Satan himself was chasing him... I stood in complete shock, still expecting to hear his steps, but nothing happened. Gradually coming to my senses, and finally managing to relax my stiff body, I took a deep breath and... lost consciousness. I woke up on the bed, drinking hot wine from the hands of my dear maid Kei. But immediately, remembering what had happened, she jumped to her feet and began to rush around the room, not having any idea what to do... Time passed, and she had to do something, come up with something in order to somehow protect herself and your family from this two-legged monster. I knew for sure that now all the games were over, that the war had begun. But our forces, to my great regret, were very, very unequal... Naturally, I could defeat him in my own way... I could even simply stop his bloodthirsty heart. And all these horrors would end immediately. But the fact is that, even at thirty-six years old, I was still too pure and kind to kill... I never took a life, on the contrary, I very often gave it back. And even such a terrible person as Karaffa was, she could not yet execute...
The next morning there was a loud knock on the door. My heart has stopped. I knew - it was the Inquisition... They took me away, accusing me of “verbalism and witchcraft, stupefying honest citizens with false predictions and heresy”... That was the end.
The room they put me in was very damp and dark, but for some reason it seemed to me that I wouldn’t stay in it for long. At noon Caraffa came...
– Oh, I beg your pardon, Madonna Isidora, you were given someone else’s room. This is not for you, of course.
– What is all this game for, monsignor? – I asked, proudly (as it seemed to me), raising my head. “I would prefer simply the truth, and I would like to know what I am really accused of.” My family, as you know, is very respected and loved in Venice, and it would be better for you if the accusations were based on truth.
Caraffa would never know how much effort it took me to look proud then!.. I understood perfectly well that hardly anyone or anything could help me. But I couldn't let him see my fear. And so she continued, trying to bring him out of that calmly ironic state, which apparently was his kind of defense. And which I absolutely couldn’t stand.
– Will you deign to tell me what my fault is, or will you leave this pleasure to your faithful “vassals”?!
“I do not advise you to boil, Madonna Isidora,” Caraffa said calmly. – As far as I know, all of your beloved Venice knows that you are a Witch. And besides, the strongest who once lived. Yes, you didn’t hide this, did you?
Suddenly I completely calmed down. Yes, it was true - I never hid my abilities... I was proud of them, like my mother. So now, in front of this crazy fanatic, will I betray my soul and renounce who I am?!
– You are right, Your Eminence, I am a Witch. But I am not from the Devil, nor from God. I am free in my soul, I KNOW... And you can never take this away from me. You can only kill me. But even then I will remain who I am... Only in that case, you will never see me again...